Using the series (see Sec.5.4.8.12 in [1]})
of the form
\( \sum ^{\infty }_{n=1}\frac{(-1)^n}{\Gamma (\nu +1+n) \Gamma (\nu +1-n)} \cos (nk)= \frac{2^{2 \nu -1}}{\Gamma (2 \nu +1)}\sin ^{2\nu } \left(\frac{k}{2}\right) -\frac{1}{2\Gamma ^2(\nu +1)} , \) 
